\(\int \frac {(a+b x^2)^{3/2} (A+B x+C x^2)}{x^2 (c+d x)^2} \, dx\) [90]

Optimal result

Integrand size = 32, antiderivative size = 359 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2 (c+d x)^2} \, dx=-\frac {b (2 c C-B d) \sqrt {a+b x^2}}{d^3}-\frac {\left (b c^2 \left (c^2 C-B c d+A d^2\right )+a d^2 \left (c^2 C-B c d+2 A d^2\right )\right ) \sqrt {a+b x^2}}{c^2 d^4 x}+\frac {b C x \sqrt {a+b x^2}}{2 d^2}+\frac {\left (b c^2+a d^2\right ) \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2}}{c d^4 x (c+d x)}+\frac {\sqrt {b} \left (3 a C d^2+2 b \left (3 c^2 C-2 B c d+A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 d^4}+\frac {\sqrt {b c^2+a d^2} \left (a d^3 (B c-2 A d)+b c^2 \left (3 c^2 C-2 B c d+A d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^3 d^4}-\frac {a^{3/2} (B c-2 A d) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^3} \] Output:

-b*(-B*d+2*C*c)*(b*x^2+a)^(1/2)/d^3-(b*c^2*(A*d^2-B*c*d+C*c^2)+a*d^2*(2*A* 
d^2-B*c*d+C*c^2))*(b*x^2+a)^(1/2)/c^2/d^4/x+1/2*b*C*x*(b*x^2+a)^(1/2)/d^2+ 
(a*d^2+b*c^2)*(A*d^2-B*c*d+C*c^2)*(b*x^2+a)^(1/2)/c/d^4/x/(d*x+c)+1/2*b^(1 
/2)*(3*a*C*d^2+2*b*(A*d^2-2*B*c*d+3*C*c^2))*arctanh(b^(1/2)*x/(b*x^2+a)^(1 
/2))/d^4+(a*d^2+b*c^2)^(1/2)*(a*d^3*(-2*A*d+B*c)+b*c^2*(A*d^2-2*B*c*d+3*C* 
c^2))*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/c^3/d^4-a^ 
(3/2)*(-2*A*d+B*c)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/c^3
 

Mathematica [A] (verified)

Time = 2.02 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2 (c+d x)^2} \, dx=-\frac {\sqrt {a+b x^2} \left (2 a d^2 (c (c C-B d) x+A d (c+2 d x))+b c^2 x \left (6 c^2 C+c (-4 B d+3 C d x)+d^2 (2 A-x (2 B+C x))\right )\right )}{2 c^2 d^3 x (c+d x)}-\frac {2 \sqrt {-b c^2-a d^2} \left (a d^3 (B c-2 A d)+b c^2 \left (3 c^2 C-2 B c d+A d^2\right )\right ) \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{c^3 d^4}+\frac {\sqrt {b} \left (3 a C d^2+2 b \left (3 c^2 C-2 B c d+A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{d^4}-\frac {a^{3/2} (B c-2 A d) \log (x)}{c^3}+\frac {a^{3/2} (B c-2 A d) \log \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c^3} \] Input:

Integrate[((a + b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^2*(c + d*x)^2),x]
 

Output:

-1/2*(Sqrt[a + b*x^2]*(2*a*d^2*(c*(c*C - B*d)*x + A*d*(c + 2*d*x)) + b*c^2 
*x*(6*c^2*C + c*(-4*B*d + 3*C*d*x) + d^2*(2*A - x*(2*B + C*x)))))/(c^2*d^3 
*x*(c + d*x)) - (2*Sqrt[-(b*c^2) - a*d^2]*(a*d^3*(B*c - 2*A*d) + b*c^2*(3* 
c^2*C - 2*B*c*d + A*d^2))*ArcTan[(Sqrt[-(b*c^2) - a*d^2]*x)/(Sqrt[a]*(c + 
d*x) - c*Sqrt[a + b*x^2])])/(c^3*d^4) + (Sqrt[b]*(3*a*C*d^2 + 2*b*(3*c^2*C 
 - 2*B*c*d + A*d^2))*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])])/d^ 
4 - (a^(3/2)*(B*c - 2*A*d)*Log[x])/c^3 + (a^(3/2)*(B*c - 2*A*d)*Log[-Sqrt[ 
a] + Sqrt[a + b*x^2]])/c^3
 

Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2353, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2 (c+d x)^2} \, dx\)

\(\Big \downarrow \) 2353

\(\displaystyle \int \left (\frac {\left (a+b x^2\right )^{3/2} (B c-2 A d)}{c^3 x}-\frac {d \left (a+b x^2\right )^{3/2} (B c-2 A d)}{c^3 (c+d x)}+\frac {\left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{c^2 (c+d x)^2}+\frac {A \left (a+b x^2\right )^{3/2}}{c^2 x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) (B c-2 A d)}{c^3}+\frac {3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+2 b c^2\right ) \left (A d^2-B c d+c^2 C\right )}{2 c^2 d^4}+\frac {3 b \sqrt {a d^2+b c^2} \left (A d^2-B c d+c^2 C\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c d^4}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a d^2+2 b c^2\right ) (B c-2 A d)}{2 c^2 d^3}+\frac {\left (a d^2+b c^2\right )^{3/2} (B c-2 A d) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^3 d^3}+\frac {3 a A \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 c^2}+\frac {a \sqrt {a+b x^2} (B c-2 A d)}{c^3}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{c^2 d (c+d x)}-\frac {3 b \sqrt {a+b x^2} (2 c-d x) \left (A d^2-B c d+c^2 C\right )}{2 c^2 d^3}-\frac {\sqrt {a+b x^2} (B c-2 A d) \left (2 \left (a d^2+b c^2\right )-b c d x\right )}{2 c^3 d^2}-\frac {A \left (a+b x^2\right )^{3/2}}{c^2 x}+\frac {3 A b x \sqrt {a+b x^2}}{2 c^2}\)

Input:

Int[((a + b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^2*(c + d*x)^2),x]
 

Output:

(a*(B*c - 2*A*d)*Sqrt[a + b*x^2])/c^3 + (3*A*b*x*Sqrt[a + b*x^2])/(2*c^2) 
- (3*b*(c^2*C - B*c*d + A*d^2)*(2*c - d*x)*Sqrt[a + b*x^2])/(2*c^2*d^3) - 
((B*c - 2*A*d)*(2*(b*c^2 + a*d^2) - b*c*d*x)*Sqrt[a + b*x^2])/(2*c^3*d^2) 
- (A*(a + b*x^2)^(3/2))/(c^2*x) - ((c^2*C - B*c*d + A*d^2)*(a + b*x^2)^(3/ 
2))/(c^2*d*(c + d*x)) + (3*a*A*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2] 
])/(2*c^2) + (Sqrt[b]*(B*c - 2*A*d)*(2*b*c^2 + 3*a*d^2)*ArcTanh[(Sqrt[b]*x 
)/Sqrt[a + b*x^2]])/(2*c^2*d^3) + (3*Sqrt[b]*(2*b*c^2 + a*d^2)*(c^2*C - B* 
c*d + A*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*c^2*d^4) + ((B*c - 2 
*A*d)*(b*c^2 + a*d^2)^(3/2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqr 
t[a + b*x^2])])/(c^3*d^3) + (3*b*Sqrt[b*c^2 + a*d^2]*(c^2*C - B*c*d + A*d^ 
2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(c*d^4) - 
 (a^(3/2)*(B*c - 2*A*d)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/c^3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2353

Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) 
^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer 
Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(743\) vs. \(2(329)=658\).

Time = 0.33 (sec) , antiderivative size = 744, normalized size of antiderivative = 2.07

method	result	size
risch	\(-\frac {a A \sqrt {b \,x^{2}+a}}{c^{2} x}+\frac {\frac {\left (A \,a^{2} d^{6}+2 A a b \,c^{2} d^{4}+A \,b^{2} c^{4} d^{2}-B \,a^{2} c \,d^{5}-2 B a b \,c^{3} d^{3}-c^{5} B \,b^{2} d +C \,a^{2} c^{2} d^{4}+2 C a b \,c^{4} d^{2}+c^{6} C \,b^{2}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{6}}+\frac {b \,c^{2} \left (d \left (B d -2 C c \right ) \sqrt {b \,x^{2}+a}+A \sqrt {b}\, d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+C b \,d^{2} \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+3 C \sqrt {b}\, c^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+\frac {2 a C \,d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-2 B \sqrt {b}\, c d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )\right )}{d^{4}}-\frac {\left (2 A \,a^{2} d^{6}-2 A \,b^{2} c^{4} d^{2}-B \,a^{2} c \,d^{5}+2 B a b \,c^{3} d^{3}+3 c^{5} B \,b^{2} d -4 C a b \,c^{4} d^{2}-4 c^{6} C \,b^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{5} c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {a^{\frac {3}{2}} \left (2 A d -B c \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c}}{c^{2}}\)	\(744\)
default	\(\text {Expression too large to display}\)	\(1542\)

Input:

int((b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^2/(d*x+c)^2,x,method=_RETURNVERBOSE)
 

Output:

-a*A/c^2*(b*x^2+a)^(1/2)/x+1/c^2*(1/d^6*(A*a^2*d^6+2*A*a*b*c^2*d^4+A*b^2*c 
^4*d^2-B*a^2*c*d^5-2*B*a*b*c^3*d^3-B*b^2*c^5*d+C*a^2*c^2*d^4+2*C*a*b*c^4*d 
^2+C*b^2*c^6)*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+( 
a*d^2+b*c^2)/d^2)^(1/2)-b*c*d/(a*d^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln(( 
2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d) 
^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))+b*c^2/d^4*(d*(B*d-2 
*C*c)*(b*x^2+a)^(1/2)+A*b^(1/2)*d^2*ln(b^(1/2)*x+(b*x^2+a)^(1/2))+C*b*d^2* 
(1/2*x/b*(b*x^2+a)^(1/2)-1/2*a/b^(3/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))+3*C* 
b^(1/2)*c^2*ln(b^(1/2)*x+(b*x^2+a)^(1/2))+2*a*C*d^2*ln(b^(1/2)*x+(b*x^2+a) 
^(1/2))/b^(1/2)-2*B*b^(1/2)*c*d*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))-1/d^5*(2*A* 
a^2*d^6-2*A*b^2*c^4*d^2-B*a^2*c*d^5+2*B*a*b*c^3*d^3+3*B*b^2*c^5*d-4*C*a*b* 
c^4*d^2-4*C*b^2*c^6)/c/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2 
*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a 
*d^2+b*c^2)/d^2)^(1/2))/(x+c/d))+a^(3/2)/c*(2*A*d-B*c)*ln((2*a+2*a^(1/2)*( 
b*x^2+a)^(1/2))/x))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2 (c+d x)^2} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^2/(d*x+c)^2,x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2 (c+d x)^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2}\right )}{x^{2} \left (c + d x\right )^{2}}\, dx \] Input:

integrate((b*x**2+a)**(3/2)*(C*x**2+B*x+A)/x**2/(d*x+c)**2,x)
 

Output:

Integral((a + b*x**2)**(3/2)*(A + B*x + C*x**2)/(x**2*(c + d*x)**2), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2 (c+d x)^2} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{2} x^{2}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^2/(d*x+c)^2,x, algorithm="maxima 
")
 

Output:

integrate((C*x^2 + B*x + A)*(b*x^2 + a)^(3/2)/((d*x + c)^2*x^2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2 (c+d x)^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^2/(d*x+c)^2,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Polynomial exponent overflow. Error 
: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2 (c+d x)^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (C\,x^2+B\,x+A\right )}{x^2\,{\left (c+d\,x\right )}^2} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^2*(c + d*x)^2),x)
 

Output:

int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^2*(c + d*x)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1680, normalized size of antiderivative = 4.68 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2 (c+d x)^2} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^2/(d*x+c)^2,x)
 

Output:

(8*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d 
+ b*c*x)*a**2*c*d**4*x + 8*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt 
(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*d**5*x**2 - 4*sqrt(a*d**2 + b*c**2)* 
log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c**3*d**2*x 
- 4*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d 
 + b*c*x)*a*b*c**2*d**3*x**2 - 4*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2 
)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c**2*d**3*x - 4*sqrt(a*d**2 + b 
*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c*d** 
4*x**2 + 8*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2 
) - a*d + b*c*x)*b**2*c**4*d*x + 8*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x* 
*2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**2*c**3*d**2*x**2 - 12*sqrt(a*d 
**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b* 
c**6*x - 12*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c** 
2) - a*d + b*c*x)*b*c**5*d*x**2 - 8*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a** 
2*c*d**4*x - 8*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*d**5*x**2 + 4*sqrt( 
a*d**2 + b*c**2)*log(c + d*x)*a*b*c**3*d**2*x + 4*sqrt(a*d**2 + b*c**2)*lo 
g(c + d*x)*a*b*c**2*d**3*x**2 + 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c 
**2*d**3*x + 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c*d**4*x**2 - 8*sqrt 
(a*d**2 + b*c**2)*log(c + d*x)*b**2*c**4*d*x - 8*sqrt(a*d**2 + b*c**2)*log 
(c + d*x)*b**2*c**3*d**2*x**2 + 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b...